Let $f(x)=a_ix^i + a_{i-1}x^{i-1} + \ldots + a_0$ where $a_i>0$. How can I proof that this general polynomial with real coeficients is $O(x^i)$ using the Big-O notation theory.
Try 1: I thought of choosing some $k=1$, and assuming that $x>1$, and then derive a $C$ such that $\frac{f(x)}{g(x)} \leq \frac{Cg(x)}{g(x)} = C$. However, I don't clrearly understand how to do it the general polynomial with real coeficients.
Consider, for $x\ge1$
$$|f(x)|=|a_ix^i+a_{i-1}x^{i-1}+\cdots a_0|\le|a_i|x^i+|a_{i-1}|x^{i-1}+\cdots |a_0|\\ \le|a_i|x^i+|a_{i-1}|x^i+\cdots|a_0|x^i$$
as $x^i\ge x^{i-1}\ge\cdots x^0$.
Then you have
$$C=|a_i|+|a_{i-1}|+\cdots|a_0|.$$